• homographic exponential models;
  • macromolecular refinement;
  • probability distributions.

Until modelling is complete, macromolecular structures are refined in the absence of a model for some of the atoms in the crystal. Techniques for defining positional probability distributions of atoms, and using them to model the missing part of a macromolecular crystal structure and the bulk solvent, are described. The starting information may consist of either a tentative structural model for the missing atoms or an electron-density map. During structure completion and refinement, the use of probability distributions enables the retention of low-resolution phase information while avoiding premature commitment to uncertain higher resolution features. Homographic exponential modelling is proposed as a flexible, compact and robust parametrization that proves to be superior to a traditional Fourier expansion in approximating a model protein envelope. The homographic ­exponential model also has potential applications to ab initio phasing of Fourier amplitudes associated with macro­molecular envelopes.